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A001329 Number of nonisomorphic groupoids with n elements.
(Formerly M4760 N2035)
52
1, 1, 10, 3330, 178981952, 2483527537094825, 14325590003318891522275680, 50976900301814584087291487087214170039, 155682086691137947272042502251643461917498835481022016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The number of isomorphism classes of closed binary operations on a set of order n.

The term "magma" is also used as an alternative for "groupoid" since the latter has a different meaning in e.g. category theory. - Joel Brennan, Jan 20 2022

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Philip Tureček, Table of n, a(n) for n = 0..27

J. Berman and S. Burris, A computer study of 3-element groupoids Lect. Notes Pure Appl. Math. 180 (1994) 379-429  MR1404949

M. A. Harrison, The number of isomorphism types of finite algebras, Proc. Amer. Math. Soc., 17 (1966), 731-737.

Eric Postpischil, Posting to sci.math newsgroup, May 21 1990

Marko Riedel Counting non-isomorphic binary relations

N. J. A. Sloane, Overview of A001329, A001423-A001428, A258719, A258720.

T. Tamura, Some contributions of computation to semigroups and groupoids, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. (Annotated and scanned copy)

Eric Weisstein's World of Mathematics, Groupoid

Wikipedia, Magma (algebra)

Index entries for sequences related to groupoids

FORMULA

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = Product_{i, j>=1} ( (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)).

a(n) is asymptotic to n^(n^2)/n! = A002489(n)/A000142(n) ~ (e*n^(n-1))^n / sqrt(2*Pi*n).

a(n) = A079173(n) + A027851(n) = A079177(n) + A079180(n).

a(n) = A079183(n) + A001425(n) = A079187(n) + A079190(n).

a(n) = A079193(n) + A079196(n) + A079199(n) + A001426(n).

MAPLE

with(numtheory);

with(group):

with(combinat):

pet_cycleind_symm :=

proc(n)

        local p, s;

        option remember;

        if n=0 then return 1; fi;

        expand(1/n*add(a[l]*pet_cycleind_symm(n-l), l=1..n));

end;

pet_flatten_term :=

proc(varp)

        local terml, d, cf, v;

        terml := [];

        cf := varp;

        for v in indets(varp) do

            d := degree(varp, v);

            terml := [op(terml), seq(v, k=1..d)];

            cf := cf/v^d;

        od;

        [cf, terml];

end;

bs_binop :=

proc(n)

        option remember;

        local dsjc, flat, p, q, len,

              cyc, cyc1, cyc2, l1, l2, res;

        if n=0 then return 1; fi;

        if n=1 then return 1; fi;

        res := 0;

        for dsjc in pet_cycleind_symm(n) do

            flat := pet_flatten_term(dsjc);

            p := 1;

            for cyc1 in flat[2] do

                l1 := op(1, cyc1);

                for cyc2 in flat[2] do

                    l2 := op(1, cyc2);

                    len := lcm(l1, l2); q := 0;

                    for cyc in flat[2] do

                        if len mod op(1, cyc) = 0 then

                           q := q  + op(1, cyc);

                        fi;

                    od;

                    p := p * q^(l1*l2/len);

                od;

            od;

            res := res + p*flat[1];

        od;

        res;

end;

----- 2nd program -----

Z := proc(n) Z(n) := 1/n*expand(add(a[k]*Z(n-k), k=1..n)) end proc;

Z(0) := 1;

magmas := proc(n)

#option remember;

uses NumberTheory;

if n=0 or n=1 then 1 else

q := 0;

for m in Z(n) do

r := 1;

S := indets(m);

for u in indets(m) do

i := op(1, u);

j := degree(m, u);

D := 0;

for d in Divisors(i) do D := D + d*degree(m, a[d]) od;

r := r*D^(i*j^2);

S := S minus {u};

for v in S do

k := op(1, v);

l := degree(m, v);

D := 0;

for d in Divisors(ilcm(i, k)) do D := D + d*degree(m, a[d]) od;

r := r*D^(igcd(i, k)*j*l*2);

od;

od;

q := q + coeffs(m)*r;

od;

fi;

end proc;

# Philip Turecek, Jul 11 2022

MATHEMATICA

magmas[n_] := (

    rul1 = {{a[i_], j_}, {a[k_], l_}} :> sum[i, k]^(j*l*GCD[i, k]*(2-Boole[i==k]));

    rul2 = {a[r_], s_} :> If[Mod[lcm, r]==0, r*s, 0];

    trans[mo_] := (

        listc = FactorList@mo;

        list = listc[[2;; ]];

        sum[i_, k_] := (

            lcm = LCM[i, k];

            Plus@@(list/.rul2)

        );

        pairs = Select[Tuples[list, 2], OrderedQ];

        listc[[1, 1]]^listc[[1, 2]]*Times@@(pairs/.rul1)

    );

    trans/@CycleIndexPolynomial[SymmetricGroup@n, Array[a, n]]

);

(* Philip Turecek, May 25 2022 *)

PROG

(Sage)

def magmas(n):

    def trans(p):

        q = 0

        for m in p.support():

            m = list(m)

            r = 1

            S = set(m).copy()

            for i in set(m):

                j = m.count(i)

                for k in S:

                    l = m.count(k)

                    D = [d*m.count(d) for d in lcm(i, k).divisors()]

                    r *= sum(D)^(gcd(i, k)*j*l*(2-(i==k)))

                S.remove(i)

            q += p.coefficients()[p.support().index(m)]*r

        return q

    return trans(SymmetricGroup(n).cycle_index())

# Philip Turecek, May 24 2022

CROSSREFS

Cf. A001424, A001425, A002489, A006448, A029850, A030245-A030265, A030271, A038015-A038023.

Sequence in context: A243008 A133198 A292443 * A007101 A007103 A006903

Adjacent sequences:  A001326 A001327 A001328 * A001330 A001331 A001332

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Formula and more terms from Christian G. Bower, May 08 1998, Dec 03 2003

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)